Alexander Iakovlev: Lower bounds for the weak type (1,1) estimate for the maximal function associated to cubes in high dimensions

In this paper we will provide the quantitative estimation for the dependence of a lower bound of the Hardy-Littlewood maximal function. This work was inspired by the paper of Stein and Strömberg where general properties of the maximal function was studied. In that work the increase with the number of dimensions d of the constant A_d that appears in the weak type (1,1) inequality for the maximal function was proved however no estimation were given. In a recent paper J.M. Aldaz shows that the lowest constant A_d tends to infinity as the dimension d tends to infinity In this paper we prove the result of J.M. Aldaz providing quantitative estimation of A_d >= Cd^{1/4} with some constant C independent of d.